The count sees the twist first

The round trip from a head’s spiral counts back to its divergence angle has survived everything done to it so far. Recovering the angle built it: count the two shortest families in a band of the head, forget the angle, and recover the interval of divergences that makes those two families shortest. A head displaced before it is counted moved every organ independently by up to two and a half spacings and found the recovery coarsening and refusing but rarely wrong. A twist is a divergence moved the head by smooth fields and found one that fools it: a twist, each organ turned about the centre in proportion to its radius, which changes the angle between one organ and the next and is recovered as exactly that — a changed angle.

That essay proposed a check with an answer on paper. The twist adds $a/(2Rr)$ radians to the divergence at radius $r$ on a head of radius $R$ twisted by $a$ at the rim, so the extra angle falls as one over the radius. An inner annulus and an outer annulus should recover different angles, and a head large enough for their intervals to be narrow should show the difference before the twist has misled the ordinary reading. The question was the head size at which that happens.

Three bands, three heads #

The heads are Vogel’s rule at the golden angle, of 900, 2,400 and 9,000 organs, twisted by nought to eight radians at the rim, read clean and with every organ also displaced independently by a quarter of a spacing, at three seeds. Each is read three times: in the single band from 0.55 to 0.95 of its radius, as the twist essay read it, and in two annuli, 0.35 to 0.6 and 0.7 to 0.95.

Two things in the reading had to change for the larger heads, and both are limits of the instrument rather than choices. A 2,400-organ head counts 89 and 144 at its rim and a 9,000-organ head 144 and 233, and the counter’s scan of index lags stopped at 90; it now runs to 400. And the recovery refused every pair containing 144, because it only considered offsets to 120, and it located the edge of an interval on a grid of a hundredth of a degree, wider than the whole band 144 and 233 allow. It now considers offsets to 400 and refines each edge by bisection to a millionth of a degree. On an untwisted 900-organ head, in all three bands, it returns the old recovery’s intervals to within the old grid.

What each band counts #

Untwisted, every band counts two consecutive Fibonacci numbers. The 900-organ head’s inner annulus counts 34 and 55, its outer annulus and single band 55 and 89. The 2,400-organ head counts 55 and 89 inside and 89 and 144 outside; the 9,000-organ head 89 and 144, and 144 and 233. Every interval holds the golden angle, 137.5078°, and they narrow with the head: the single band’s is 0.087° wide on 900 organs, 0.033° on 2,400 and 0.010° on 9,000.

Twisted by half a radian, the 900-organ head’s outer annulus and single band count 34 and 89. Both are Fibonacci numbers, and they are not consecutive: 55 is missing between them. The recovered interval, 137.478° to 137.580°, still holds the golden angle.

Read across all three heads, the counts go wrong in a fixed order as the twist grows. First a band skips a Fibonacci number — 34 and 89, or 89 and 233. Then bands begin to count pairs sharing a factor, 34 and 68 or 89 and 178, which the recovery refuses as a whorled pattern. Only later does any band return a pair whose interval excludes the golden angle, and by then its pair is usually not two Fibonacci numbers at all — 34 and 81, 89 and 123, 13 and 47. The 9,000-organ head’s outer annulus is the exception, and a telling one: at three quarters of a radian it counts 89 and 233 and recovers 137.5080° to 137.5179°, which misses the golden angle by two ten-thousandths of a degree. The reading that misleads it is the skipped pair itself.

Where the head stops being countable #

Between the flag and the misleading reading there is a stretch where the twisted head cannot be counted at all. At two radians every band of the 900-organ head counts 34 and 68: the spirals of the 34-family have been wound so far that each pair of them reads as one family of 68, and a pair that shares a factor is refused as a whorled pattern. A person looking at the head would see the same thing the counter does — spirals that come in pairs, a head that looks as though it grew two organs at a time.

By six radians the head counts again, and wrongly. The inner annulus reads 13 and 47 and the outer 34 and 81, pairs made of one Fibonacci number and one number from no sequence the head was built on, and the single band recovers 137.699° to 137.832°, a fifth of a degree above the truth. The picture is still a head with spirals in it, and nothing about it looks disordered; the spirals are simply the ones a head at a larger divergence would show, turned further at the rim than at the centre. That is the twist the round trip was first found to be fooled by, and by the time a head looks like this the count has been flagging it for five and a half radians.

The annuli never catch it #

The two annuli’s intervals separate — the inner and outer recover angles that cannot both be right — at a twist of eight radians on the 900-organ head, at no twist read on the 2,400-organ head, and on the 9,000-organ head only when it is also displaced, again at eight. The single band has been misled by then in every case: from six radians on 900 organs, and from two on 2,400 and 9,000. The earlier essay’s paper estimate put the 900-organ separation at about six radians; measured, the intervals still overlap there, the inner annulus’s 13 and 47 allowing 137.70° to 138.05° and the outer’s 34 and 81 137.72° to 137.82°.

The reason the annuli fail is visible on the 9,000-organ head. As the twist grows the annuli do not keep counting pairs whose intervals drift apart. They stop counting usable pairs. From a radian and a half the inner annulus counts 89 and 178 and refuses; from three radians both annuli do. A check that needs two recovered intervals is silent exactly when it is needed, and when the annuli recover again — the inner at six radians, both at eight — they are already wrong.

The 2,400-organ head, band by band #

The middle head shows the one change a twist makes that is harmless. At half a radian its single band stops counting 89 and 144 and counts 55 and 89 instead — two consecutive Fibonacci numbers, a different pair — and recovers 137.486° to 137.527°, which holds the golden angle. The twist has moved a transition out of the band rather than into it, so the band reads the pair that lies wholly inside it, and nothing about the reading is wrong. A count that changes to another consecutive pair is not a flag, and the flag rightly ignores it.

The harmful change follows at three quarters of a radian, when the inner annulus counts 34 and 89 and the outer 89 and 178, and from there the head’s bands spend most twists refusing. When they recover again they have lost the sequence: the single band’s 89 and 123 at two radians, 137.536° to 137.567°, the outer annulus’s 34 and 123 at four, and 34 and 157 at six — none of them Fibonacci pairs, all of them excluding the golden angle, and not one of them recovered by both annuli at once.

Displacement makes the largest head refuse #

Adding independent displacement of a quarter of a spacing changes one thing, on one head. The 900- and 2,400-organ heads flag and are misled at the same twists displaced as clean, on all three seeds. The 9,000-organ head, displaced, still flags at three quarters of a radian — but its single band is not misled until eight radians, against two when clean. At two radians it still recovers the golden angle, and from three to six it refuses.

So displacement, which the displaced round trip found coarsens the recovery and makes it refuse rather than mislead, does the same for a twisted head: at nine thousand organs the scatter it adds is enough to break the narrow skipped pairs into refusals. It helps in the way that a refusal helps — the reader is not told anything false — and the flag has already been raised before either happens.

Larger heads are misled sooner #

The expectation was that a larger head, with its narrower intervals, would catch a twist sooner. The twist’s own arithmetic says the size should not matter either way. At the middle of the single band a twist of one radian shifts the divergence by 0.042° on the 900-organ head, 0.016° on 2,400 and 0.0042° on 9,000 — falling in proportion to the organ count — and the single band’s interval shrinks nearly as fast, from 0.087° to 0.033° to 0.010°. The shift is about half the interval on every head.

What differs is the count. On the 900-organ head the first wrong count, 34 and 89, allows a wide band of angles, 0.12° across, which holds the golden angle until the twist is large. On the larger heads the skipped pairs are higher — 89 and 233 — and their bands are narrow, 0.01°, so a small shift is enough to push the recovered interval off the truth. So the 2,400- and 9,000-organ heads are misled from two radians and the 900-organ head only from six: a larger head is not a more sensitive check on a twist, it is a more precise reading of a twisted angle.

The count flags it, every time #

The check that works is not a comparison of two recoveries. It is the count, read for what it is before anything is recovered from it.

At half a radian on the 900-organ head, and at three quarters on the 2,400- and 9,000-organ heads, some band’s count stops being two consecutive Fibonacci numbers — on every seed, clean and displaced alike. That is a twelfth of the twist that misleads the 900-organ head and three eighths of the twist that misleads the larger ones. And no untwisted head ever raises the flag: across every head, clean or displaced by a quarter of a spacing, and every twist up to a quarter of a radian, every band counts two consecutive Fibonacci numbers.

The flag is not a new instrument. A head displaced before it is counted sorted its displaced readings into the head’s own pair, a neighbour on its sequence, two members of its sequence that are not consecutive, a pair off it, and refusals. The twist moves a count into the third class — spanning, skipping a term — at displacements where independent error never did. What the two-annulus check was supposed to detect from two recovered angles, the counter detects from one pair of integers.

Why a twist makes a count skip #

The count skips because a twist moves the head’s transitions. The band essay measured it directly: under a twist each flip ring — where one family hands over to the next — moves to where the twisted divergence puts it, successive rings in opposite directions. An annulus drawn at fixed radii then straddles a transition that the untwisted head kept outside it, and a counter reading the two shortest families across a moved transition sees one family from each side of it: 34 from inside and 89 from outside, with the 55 between them never the second shortest anywhere in the band.

That is why the flag appears at such small twists and why it does not depend much on the head. It needs only one transition moved into a band, and every head has transitions within a fraction of a spacing of its bands’ edges. The counts change with radius on every head; a twist makes them change at the wrong radius, and a pair that skips a Fibonacci number is what a counter reports when a band holds a transition it was not placed across.

What this does to the round trip #

What a count is worth established that a reported pair pins the divergence to a band about $221°/mn$ wide, and a count that can be wrong by one that a wrong count usually announces itself by sharing a factor. The twist adds a third way to be wrong, and it announces itself too, differently: the pair is coprime, it is two Fibonacci numbers, and it skips one. A reader who recovers an angle only from consecutive Fibonacci pairs, and treats a skipped pair as a flag rather than a reading, is never misled by any twist read here, on any head.

The cost is refusals. From half a radian to eight on the 900-organ head, and from three quarters on the larger heads, such a reader recovers nothing from at least one band. That is the honest answer for a head whose divergence is not one number across its face: a twisted head has no single divergence to recover, and a reading that refuses to give it one is right.

The second annulus is still worth reading #

The annuli failed as a comparison of angles, but reading two of them still helps, because the flag is more likely to show in two bands than in one. On the 2,400-organ head at three quarters of a radian the single band still counts 55 and 89 and recovers the golden angle; it is the inner annulus, counting 34 and 89, and the outer, counting 89 and 178, that raise the flag. A reader with only the single band would have seen nothing wrong until a radian. Counted in two annuli, the head gives itself away a quarter of a radian sooner.

What these heads leave out #

The twist here is rigid: every organ turned by exactly $a\,r/R$. A real head’s twist, if it has one, comes from uneven growth and need not be proportional to radius, and the flag’s timing depends on which transitions it moves into which bands.

The counts are exact. A real counter can miscount by one, and a miscount can also produce a pair that is not consecutive; the flag then has two possible causes, and telling them apart needs a second band or a second count of the same band.

And the heads are golden. The Lucas branch has its own sequence, and the flag there is a pair that skips a Lucas number; the arithmetic carries over and the measurement was not repeated.

Findings that would overturn it #

An untwisted head, clean or displaced by a quarter of a spacing, with any band counting a pair that is not two consecutive Fibonacci numbers. A twist that misleads some band before any band’s count skips. A head size at which the two annuli’s intervals separate before the single band is misled.

Still open: a twist that is not proportional to radius #

A head twisted by uneven growth turns its organs by an amount that need not grow in proportion to radius; it could be concentrated near the rim, or near the centre. The measurement is the same three bands under twists of the form $a (r/R)^k$ for $k$ from a half to three, asking whether the flag still comes before the misleading reading when the twist is concentrated where the counts are high, and whether a twist confined to the centre — which moves the inner transitions and leaves the outer ones — can mislead the single band without any band’s count skipping.